We show that the limit set of an isometric and convex cocompact action of a surface group on a proper geodesic hyperbolic metric space, when equipped with a visual metric, is a Falconer–Marsh (weak) quasicircle. As a consequence, the Hausdorff dimension of such a limit set determines its bi-Lipschitz class. We give applications, including the existence of almost-isometries between periodic negatively curved metrics on ℍ2 that cannot be realized equivariantly.